The Championship side agreed with Butland earlier in the summer that he can leave for the Premier League if potential suitors make an offer of £20m plus add-ons.
According to the Daily Mail, Bournemouth must now decide whether to return with an improved bid after Stoke "flatly rejected" their opening £12m offer.
Butland, who joined Stoke in 2013, has made nine appearances for England and remained in Gareth Southgate's squad last season despite playing in the Championship.
Stoke, while happy to keep Butland, will listen to offers for a number of outfield players - including £6m midfielder Ryan Woods and former Barcelona forward Bojan Krkic - as they look to address their bloated squad this summer. body check tags ::